Bridge structures tend to vibrate under loads from the outside wind and vehicles, resulting in discomfort vehicles and pedestrians, accumulating fatigue damages to and shorter lifetime of the bridge even collapses of the bridge structures. For example, Tacoma Narrows Bridge in the United States collapsed under the induction of wind loads. Whether a bridge will vibrate sharply or not relates to the ratio of the induction frequency of external loads and the natural vibration frequency of the bridge. When these two frequencies are close to each other, the bridge structure will resonate in the mode corresponding to the natural vibration frequency. The tuned mass damper (TMD) functions to adjust the damper frequency to make it close to the controlled frequency of the bridge structure (i.e. the vibration frequency to put the bridge under control), enable the active mass in the damper to vibrate at a larger amplitude than the displacement of the bridge structure to transfer its vibration energy, while dissipate the energy transmitted to the damper by its interior damping energy dissipation device for the ultimate purpose of suppressing the vibration of the bridge structure. For the controlled mode of the bridge structure, the tuned damper increases its modal damping to maintain a low amplitude vibratory response to continuous external inductions while make the structural vibratory response can be attenuate rapidly after external inductions disappear.
As the bridge span increases, the natural vibration frequency of the bridge structure gradually decreases. For example, the controlled frequency of non-navigable beam bridge of Hong Kong-Zhuhai-Macao Bridge is 0.33 Hz approximately. However, the main beam of the suspension bridge or the cable-stayed bridge has a lower vertical vibration frequency; Xihoumen Bridge for example, with level 1 to 10 vertical vortex frequency of the main beam ranging from 0.079 Hz to 0.374 Hz. And based on the on-site measurements, the vortex induced vibration of the main beam has a frequency around 0.23 Hz to 0.32 Hz even if the beam is under a high-level vertical bending mode. All these vibrations can be effectively suppressed by using the TMD theory. But for the damper to exert a tuned damping effect, the damper frequency must be as low as the frequency just mentioned.
The main challenges for the conventional TMD to achieve ULF are insufficient stiffness of springs, along with the dead weight of the mass block which is hard to balance. If the gravity of the mass block is directly balanced by the spring elasticity only, the static elongation of the spring shall be:
                    δ        =                              mg            k                    =                                    mg                                                m                  ⁡                                      (                                          2                      ⁢                      π                      ⁢                                                                                          ⁢                      f                                        )                                                  2                                      =                          g                                                (                                      2                    ⁢                    π                    ⁢                                                                                  ⁢                    f                                    )                                2                                                                        (        1        )            
wherein δ is the spring static elongation (in m); m, the mass of the mass block (in kg); k, the spring stiffness (in N/m); g, the gravitational acceleration (in m/s2); and f, the damper frequency (in Hz). The formula above indicates that the spring static elongation (δ) is inversely proportional to the squared frequency (f), and as the frequency decreases, the spring static elongation increases rapidly, so does the space occupied by the vibration absorber and its mass. To achieve such elongation and ensure the stress of the spring wire conforms to the requirements of design specifications, the diameter and the total length of the spring wire shall also increase rapidly, raising the spring's mass in no time. The table below shows the corresponding relations among the frequency, the spring static deformation and the spring mass ratio (spring mass/block mass) when the damper is under ULF condition.
TABLE 1Corresponding relations among frequency, spring staticdeformation and spring mass ratio when conventionalTMD is under ULFConventionalSpring staticTMD frequencyelongationSpring mass/(Hz)(m)block mass0.50.9919%0.41.5524%0.32.7652%0.26.21101%0.124.82250%Note:the spring design stress [σ] = 370 MPa.
According to the table above, the length and the mass of the spring will increase rapidly with the required frequency declines. For a conventional TMD, it's difficult to reach the natural vibration frequency between 0.3 Hz and 0.5 Hz, let alone a value below 0.3 Hz.
To address those issues, as shown in FIG. 1, Tokyo Bay Bridge employs a lever-type TMD to implement a damper with frequency of 0.33 Hz by utilizing the 5-time magnifying effect of the lever to reduce the spring static compression by 5 times which changes the original spring deformation from 2.28 m to 0.46 m. With this solution, the installation space can be reduced through lever switching, however disadvantages ensued are as follows:
1. The spring consumption can barely be reduced, while the damper needs a complicated lever-based transmission system with a sophisticated structure, which is highly demanding for manufacturing accuracy and field assembly. And the materials without damping effects, such as bearings, brackets, levers and hinge joints, substantially increase the cost for vibration reduction.
2. It's hard to make the frequency even lower, and considering the damper needs a complicated lever-based transmission system, more rotary hinges brings more rotational frictions from the bearings, which gives a greater initial damping to the damper and makes it insensitive to small amplitudes. And in this case, it is also hard to optimize the parameters of the damper. In conclusion, it is very difficult for such solution to be promoted and applied to structural vibration control with requirements on lower frequency.